Schedule

• Jan 21
  Tuesday, 1:30-2:30pm, 2-449

  Philip Easo (Caltech)

  Cutsets and percolation

  Abstract: The classical Peierls argument establishes that percolation on a graph G has a non-trivial (uniformly) percolating phase if G has “not too many small cutsets”. Severo, Tassion, and I have recently proved the converse. Our argument is inspired by an idea from computer science and fits on one page. Our new approach also resolves a conjecture of Babson of Benjamini from 1999 and provides a much simpler proof of the celebrated result of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that percolation on any transitive graph with superlinear growth undergoes a non-trivial phase transition.

• Feb 3

  Yi Han (MIT)

  Some progress and mysteries in the study of inhomogeneous random matrices

  Abstract: The spectrum of a random matrix is well-understood for an invariant ensemble, and sufficiently strong information has been obtained for mean field type random matrices. There are however far more general situations where much less information is obtained: when the entries have a heavy-tailed distribution or when the variance profile has a specific structure. In this talk I discuss some recent discoveries in the latter regime. The topics include: a sharp description for the edge of a symmetric random matrix when its tail decays precisely like x^{-4} (the transition regime); a very weak condition for determining spectral outliers for finite rank perturbations of non-Hermitian random matrices with a banded variance profile; the smallest singular value for rectangular random matrix with entries in the domain of attraction of alpha-stable law; and on convergence to the circular law for ESDs of some nonhomogeneous matrices. The work employs recent concentration inequalities invented by Bandeira, Boedihardjo, van Handel and Brailovskaya. While satisfying results are obtained for some problems, a sharp understanding is still not obtained for other problems despite significant quantitative improvement.

• Feb 13
  Thursday, 3pm, 2-132

  Sasha Glazman (University of Innsbruck)

  Six-vertex model in the FKG regime

  Abstract: The six-vertex model is in correspondence with graph homomorphisms from $Z^2$ to $Z$. If a face is a saddle, it receives weight $c$, otherwise it receives weight $a$ or $b$. The distribution is proportional to the product of the weights. When $c\geq a$,$b$, a positive association (FKG) inequality provides additional tools.

  We discuss two results:

  • When $a,b \leq c \leq a+b$, we give a soft purely probabilistic proof of delocalisation relying on the non-coexistence theorem of Zhang and Sheffield. The same argument applies also to random Lipschitz functions on the triangular lattice.
  • When $c > a+b$ (localised regime), we show convergence of an interface under Dobrushin boundary conditions to the Brownian bridge.

  Joint works with Dober, Lammers, Ott.

• Feb 17

  President's Day

• Feb 26
  Wednesday, 4:15pm, 2-139

  Yair Shenfeld

  The Brownian and Poisson transport maps

  Abstract: Transport maps serve as a powerful tool to transfer information from source to target measures. However, this transfer of information is possible only if the transport map is sufficiently regular, which is often difficult to show. I will explain how taking the source measure to be an infinite-dimensional measure, and building transport maps based on stochastic processes, solves some of these challenges both in the continuous and discrete settings.

• Mar 3

  Michael Salins (Boston University)

  When do SPDEs explode?

  Abstract: Classic existence and uniqueness theorems for stochastic partial differential equations (SPDEs) prove that if the forcing terms are globally Lipschitz continuous, then there exists a unique, global solution. In this talk, I describe some of the ways that superlinear deterministic and stochastic forcing terms can combine to either cause or prevent explosion.

• Mar 10

  Sofiia Dubova (Harvard)

  Gaussian statistics for left and right eigenvectors of complex non-Hermitian random matrices

  Abstract: Eigenvector statistics of Hermitian random matrix ensembles have been extensively studied in random matrix theory. The non-Hermitian version of this question introduces additional difficulties. In this talk, we consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-1+\varepsilon}$ and show that arbitrary constant rank projections of these eigenvectors are asymptotically Gaussian and jointly independent. To prove the result in the case of a Gaussian-divisible ensemble we derive explicit integral formulas for the moment generating functions of the eigenvector projections using partial Schur decomposition. We extend this to a general i.i.d. ensemble using a Green function comparison argument.

• Mar 17

  Anna Brandenberger (MIT)

  Network archaeology of random nearest neighbour trees

  Abstract: Network archaeology is the inference of the history of a large randomly grown graph. In this talk, I will introduce the root-finding problem in a geometric setting. We consider a random nearest-neighbour tree generated by sequentially embedding vertices uniformly in the d-dimensional torus, with each new vertex connecting to the nearest existing vertex. Given only the unlabeled tree and a target error ε, a root-finding algorithm identifies a small "confidence set" of vertices that contains the root with probability at least 1-ε. We define three geometric variants of this problem, differing in the amount of metric information provided in addition to the graph structure. In the case where the torus embedding is given, we show that there exist efficient root-finding algorithms, and we derive both algorithmic upper bounds and information-theoretic lower bounds on the size of the confidence set. In particular, in d=1, we prove matching bounds for a confidence set of size Θ((log(1/ε))/(log log(1/ε))).

• Mar 24

  Spring Break

• Mar 31
  3:00pm

  Oren Yakir (Stanford)

  Random roots near the unit circle

  Abstract: It is well known that a random polynomial with i.i.d. coefficients has most of its roots clustering around the unit circle as the degree grows large. In this talk, we will survey recent results on finer properties, including the typical distance between the random roots and the unit circle, and the separation distances between the roots.

  Based on joint works with Nicholas Cook, Hoi Nguyen, Marcus Michelen, and Ofer Zeitouni.

• Apr 7

  Youngtak Sohn (Brown)

  Stochastic Block Model with Many Communities

  Abstract: The stochastic block model (SBM), a random graph generalizing the Erdős–Rényi model, has long served as a framework for community detection. For SBMs with $n$ vertices and a fixed number of communities $q$, Decelle et al. (2011) predicted that efficient recovery is possible above the Kesten–Stigum (KS) threshold and impossible below it. We review recent progress toward proving this conjecture. We then turn to the case where $q = q_n$ grows with $n$, a setting for which no prediction currently exists. We show that the KS threshold can be surpassed efficiently when $q_n \gg \sqrt{n}$, while low-degree algorithms fail to beat the KS threshold when $q_n \ll \sqrt{n}$. Based on joint work with Byron Chin, Elchanan Mossel, and Alex Wein.

• Apr 14

  Zijie Zhuang (UPenn)

  Percolation of thick points of the log-correlated Gaussian field in high dimensions

  Abstract: The log-correlated Gaussian field is the higher-dimensional analog of the 2D Gaussian free field. In this talk, I will present a result showing that the set of thick points of the log-correlated Gaussian field contains an unbounded path in high dimensions. I will discuss how we get intuition from fractal percolation and the application to the exponential metric of the log-correlated Gaussian field. Based on joint work with Jian Ding and Ewain Gwynne.

• Apr 21

  Patriot's Day

• Apr 28
  3:15pm

  Arka Adhikari (University of Maryland)

  Moderate Deviations For The Capacity Of The Random Walk Range In Dimension Four

  Abstract: We find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen concerning the volume of the random walk range for d = 2. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, $\begin{equation*} \inf_{f: \|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{1/2}_{L^2} \|\nabla f\|^{1/2}_{L^2}}{ [\int_{(\mathbb{R}^4)^2} f^2(x) G(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}. \end{equation*} $

• Apr 28
  4:30pm

  Sandro Franceschi (Télécom SudParis)

  Degenerate systems of three Brownian particles with asymmetric collisions

  Abstract: We consider a degenerate system of three particles which collide asymmetrically. The system is degenerate because the particle in the middle is Brownian and the others are ballistic. We study this system's gap process and focus on its invariant measure. The gap process is an obliquely reflected degenerate Brownian motion in a quadrant. We fully characterise the cases where the Laplace transform of the invariant measure is simple, that is rational, algebraic, differentially finite or differentially algebraic. In all these cases, we determine an explicit formula for the invariant measure in terms of a Theta-like function to which we apply a (sometimes fractional) differential operator.

  To show our results, we start with a functional equation that characterises the Laplace transform of the invariant measure, derive a finite difference equation, and use Tutte’s invariant approach and some Difference Galois theory.

  This presentation is based on joint work with T. Ichiba, I. Karatzas, and K. Raschel and an upcoming work with T. Dreyfus and J. Flin.

• May 5

  Matthew Nicoletti (UC Berkeley)

• May 12

  Melanie Matchett Wood (Harvard)